Answers to Practice Questions

2.a.The
put places a floor on value of investment, i.e., less risky than buying
stock.The risk reduction comes at the
cost of the option premium.

b.Benefit from upside, but also lose on the downside.

c.A naked option position is riskier than the underlying asset. Investor gains from increase in stock price,
but loses entire investment if stock price is less than exercise price at
expiration.

d.Investor exchanges uncertain upside changes in stock price for
the known up-front income from the option premium.

e.Safe investment if the debt is risk free.

f.From put-call parity, this is equivalent (for European
options) to ‘buy bond.’Therefore, this
is a safe investment.

g.Another naked, high-risk position with known up-front income
but exposure to down movements in stock price.

3.While it is true that both the buyer of a call and the seller
of a put hope the price will rise, the two positions are not identical.The buyer of a call will find her profit
changing from zero and increasing as the stock price rises (see text Figure 20.2),
while the seller of a put will find his loss decreasing and then remaining at
zero as the stock price rises (see text Figure 20.3).

4.You would buy the American call for $75, exercise the call immediately
in order to purchase a share of Pintail stock for $50, and then sell the share
of Pintail stock for $200.The net gain
is: [$200 – ($75 + $50)] = $75.

If the call is a European call, you
should buy the call, deposit in the bank an amount equal to the present value
of the exercise price, and sell the stock short.This produces a current cash flow equal to: [$200 – $75 – ($50/1
+ r))].At the maturity of the call, the
action depends on whether the stock price is greater than or less than the exercise
price.If the stock price is greater
than $50, then you would exercise the call (using the cash from the bank
deposit) and buy back the stock.If the
stock price is less than $50, then you would let the call expire and buy back
the stock.The cash flow at maturity is
the greater of zero (if the stock price is greater than $50) or [$50 – stock
price] (if the stock price is less than $50).Therefore, the cash flows are positive now and zero or positive one year
from now.

5.Let P3 = the value of the three month put, C3
= the value of the three month call, S = the market value of a share of stock,
and EX = the exercise price of the options.Then, from put-call parity:

C3 + [EX/(1 + r)0.25] = P3 + S

Since both options have
an exercise price of $60 and both are worth $10, then:

EX/(1 + r)0.25 = S

From put-call parity for
the six-month options, we have:

C6 + [EX/(1 + r)0.50] = P6 + S

Since S = EX/(1 + r)0.25,
and EX/(1 + r)0.50 is less than EX/(1 + r)0.25, then the
value of the six-month call is greater than the value of the six-month put.

6.[Note: In the first printing of the seventh edition, the call
option price is shown incorrectly as $2.30.The price should be $12.30.]

[Note:The answer to (b) ignores dilution.Chapter 23 discusses how dilution affects the valuation of
warrants and convertibles.Dilution has
a similar effect on the valuation of standby underwriting.This is because, if the option is exercised,
the underwriter pays the issue price, but also obtains an equity stake in this
new money.After reading Chapter 23,
students might return to the issue of the effect of dilution on the value of
standby agreements.]

9.The $100 million threshold can be viewed as an exercise
price.Since she gains 20% of all
profits in excess of this level, it is comparable to a call option.Whether this provides an adequate incentive
depends on how achievable the $100 million threshold is and how Ms. Cable
evaluates her prospects of generating income greater than this amount.

10.a.The
payoffs at expiration for the two options are shown in the following position
diagram:

Taking into
account the $100 that must be repaid at expiration, the net payoffs are:

Thus, to replicate the payoffs for
the put, you would buy a 26-week call with an exercise price of $100, invest
the present value of the exercise price in a 26-week risk-free security, and
sell the stock short.

b.Using the put-call parity relationship, the European put will
sell for:

The buyer of the
straddle profits if the stock price moves substantially in either direction;
hence, the straddle is a bet on high variability.The buyer of the butterfly profits if the stock price doesn’t
move very much, and hence, this is a bet on low variability.

17.a.The
bond value increases to the present value of the guaranteed payoff, valued at
the risk-free rate:

Bond
value = ($50 + $5)/1.08 = $50.93

b.The payoffs to stockholders are unaffected.If the firm defaults, its bondholders are
paid off, but shareholders get nothing, just as before.If the firm does not default, payments to
shareholders do not change.

c.The firm effectively acquires a new asset, the government
guarantee worth $25.93 (the difference between the previous and new bond
values).The firm’s balance sheet could
be expressed this way:

Asset
value

$30.00

$50.93

Bonds

Government’s guarantee

25.93

5.00

Stock

$55.93

$55.93

Firm value

d.By issuing 10-percent bonds with $50 face value, Rectangular
raises $50.93 cash (the present value, at 8 percent, of $55), which is used to
repurchase stock.After the
transaction, the market value balance sheet is the same as Circular’s.Shareholders have pocketed the $25.93 value
of the government guarantee.

18.Answers here will vary according to the stock and the specific
options selected, but all should exhibit properties very close to those
predicted by the theory described in the chapter.

19.Imagine two stocks, each with a market price of $100.For each stock, you have an at-the-money
call option with an exercise price of $100.Stock A’s price now falls to $50 and Stock B’s rises to $150.The value of your portfolio of call options
is now:

Value

Call on A

0

Call on B

50

Total

$50

Now compare this with the value of
an at-the-money call to buy a portfolio with equal holdings of A and B.Since the average change in the prices of the
two stocks is zero, the call expires worthless.

This is an example of a general
rule: An option on a portfolio is less valuable than a portfolio of options on
the individual stocks because, in the latter case, you can choose which options
to exercise.

20.Consider each company in turn, making use of the put-call
parity relationship:

Drongo
Corp.Here, the left-hand side [52
+ (50/1.05) = 99.62] is less than the right-hand side [20 + 80 = 100].Therefore, there is a slight
mispricing.To take advantage of this
situation, one should buy the call, invest $47.62 at the risk-free rate, sell
the put, and sell the stock short.

Ragwort,
Inc.Here, the left-hand side [15 +
(100/1.05) = 110.24) is greater than the right-hand side [10 + 80 = 90].Therefore, there is a significant
mispricing.To take advantage of this
situation, one should sell the call, borrow $95.24 at the risk-free rate, buy
the put, and buy the stock.

Wombat
Corp.For the three-month option,
the left-hand side [18 + (40/1.025) = 57.02] and the
right-hand side [7 + 50 = 57] are essentially equal, so
there is no mispricing.

For the first six-month option, the
left-hand side [17 + (40/1.05) = 55.10] is slightly greater than the right-hand
side [5 + 50 = 55], so there is a slight mispricing.

For the second six-month option, the
left-hand side [10 + (50/1.05) = 57.62] is slightly less than the right-hand side
[8 + 50 = 58], and so there is a slight mispricing.

21.The value of the options increases if the variance of the cash
flows increases.Therefore, you will
prefer the riskier proposal.

22.One strategy might be to buy straddle, that is, buy a call and
a put with exercise price equal to the asset’s current price.If the asset price does not change, both
options become worthless.However, if
the price falls, the put will be valuable and, if price rises, the call will be
valuable.The larger the price movement
in either direction, the greater the profit.

If
investor’s have underestimated volatility, the option prices will be too
low.Thus, an alternative strategy is
to buy a call (or a put) and hedge against changes in the asset price by
simultaneously selling (or, in the case of the put, buying) delta shares of
stock.

Challenge Questions

1.Letter the diagrams in Figure 20.13 (a) through (d), beginning
in the upper-left corner and proceeding clockwise.Then we have the following diagram interpretations:

a.Purchase a call with a given exercise price and sell a call
with a higher exercise price; borrow the difference necessary.(This is known as a ‘Bull Spread.’)

b.Sell a put and sell a call with the same exercise price.(This is known as a ‘Short Straddle.’)

c.Buy one call with a given exercise price, sell two calls with
a higher exercise price, and buy one call with a still higher exercise
price.(This is known as a ‘Butterfly
Spread.’)

d.Borrow money and use this money to buy a put and buy the
stock.

2.a.If
the land is worth more than $110 million, Bond will exercise its call
option.If the land is worth less than
$110 million, the buyer will exercise its put option.

b.Bond
has: (1) sold a share; (2) sold a put; and (3) purchased a call.Therefore:

This is equivalent to:

c.The interest rate can be deduced using the put-call parity
relationship.We know that the call is worth
$20, the exercise price is $110, and the combination [sell share and sell put
option] is worth $110.Therefore:

d.From the answer to Part (a), we know that Bond will end up
owning the land after the expiration of the options.Thus, in an economic sense, the land has not really been sold,
and it seems misleading to declare a profit on a sale that did not really take
place.In effect, Bond has borrowed
money, not sold an asset.

3.One way to profit from Hogswill options is to purchase the
call options with exercise prices of $90 and $110, respectively, and sell two
call options with an exercise price of $100.The immediate benefit is a cash inflow of:

[(2 ´ 11)
- (5 + 15)] = $2

Immediately
prior to maturity, the value of this position and the net profit (at various
possible stock prices) is:

Stock
Price

Position
Value

Net
Profit

85

0

0
+ 2 = 2

90

0

0
+ 2 = 2

95

5

5
+ 2 = 7

100

10

10
+ 2 = 12

105

5

5
+ 2 = 7

110

0

0
+ 2 = 2

115

0

0
+ 2 = 2

Thus, no matter what the final stock
price, we can make a profit trading in these Hogswill options.

It
is possible, but very unlikely, that you can identify such opportunities from
data published in the newspaper.Someone else has most likely already noticed (even before the paper was
printed, much less distributed to you) and traded on the information; such
trading tends to eliminate these profit opportunities.